Rigidity and sensitivity on uniform spaces

By Xinxing Wu, Yang Luo, Xin Ma, Tianxiu Lu. Topology and its Applications, 2019

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Abstract In this paper, we introduce the concepts of weak uniformity, uniform rigidity and multi-sensitivity for uniform (not necessarily compact or metric) spaces and obtain some equivalent characterizations of uniform rigidity. In particular, we prove that a dynamical system ( X , f ) defined on a Hausdorff uniform space is uniformly rigid if and only if ( X , f n ) is uniformly rigid for some/all n ∈ N if and only if its hyperspatial dynamical system is uniformly rigid or weakly rigid. Besides, we show that every non-minimal point transitive dynamical system defined on a Hausdorff uniform space with dense Banach almost periodic points is sensitive and obtain the equivalence of the multi-sensitivity between original dynamical system and its hyperspatial dynamical system for Hausdorff uniform spaces.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/701807c18b67cf4001e69538e05ae62d57d49fa3